Written by the Fiveable Content Team • Last updated September 2025
Written by the Fiveable Content Team • Last updated September 2025
Definition
A smooth function is one that has continuous derivatives up to the required order over a given interval. In calculus, this typically means the function is differentiable and its derivative is also continuous.
5 Must Know Facts For Your Next Test
For a curve to be considered smooth, it must have no sharp corners or cusps within the interval of consideration.
A smooth function used in arc length calculations ensures that the integral for arc length converges properly.
Smoothness of a curve implies the existence of higher-order derivatives which are necessary for accurate surface area computations.
The concept of a smooth function can be extended to multivariable functions when computing surface area by ensuring partial derivatives are continuous.
In applications of integration, smoothness helps in simplifying the use of parametrization techniques for curves and surfaces.
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Related terms
Differentiable: A function is differentiable at a point if it has a defined derivative at that point.
Continuous Function: A function is continuous if there are no breaks, jumps, or holes in its graph.
$C^n$ Class Functions: $C^n$ class functions have continuous derivatives up to the $n^{th}$ order.